From the triangle to the trigonometric circle

Review: ratios that preserve the angle

In a right triangle, for acute angle θ, sin θ is opposite leg over hypotenuse, cos θ adjacent leg over hypotenuse and tan θ opposite over adjacent. Two right triangles with the same θ are similar, so their ratios do not depend on size. This independence allows trigonometric numbers to be assigned to the angle itself.

If the legs are 8 and 15 and hypotenuse 17, for the angle opposite side 8 we have sin θ = 8/17, cos θ = 15/17 and tan θ = 8/15. Pythagoras confirms 8² + 15² = 64 + 225 = 289 = 17². Switching angles swaps the legs' roles but does not change the hypotenuse.

The usual right triangle shows only acute angles, although a direction can rotate by 120°, by −45° or through several turns. To study those cases we need a model allowing negative coordinates and directed rotations. That model is the radius-1 circle centered at the origin.

Rotations and the definition of a radian

A directed angle starts on an initial ray and ends on another after a rotation. In the Cartesian plane, we start on the positive x-axis. Counterclockwise is positive; clockwise negative. A complete turn is 360°, but division into degrees is a human convention.

The radian comes from geometry: if an arc of length s on a radius-r circle subtends central angle θ, its radian measure is defined as θ = s/r. Since both lengths use the same unit, the quotient is dimensionless. We can write “rad” to recall the angular convention, especially in physical applications.

One turn has length 2πr, so it measures 2π rad. Thus 180° = π rad and 90° = π/2 rad. To convert degrees to radians multiply by π/180; the reverse uses 180/π. The formula s = rθ requires θ in radians. Using a numerical degree value in that formula without conversion gives the wrong scale.

Coordinates as sine and cosine

On the unit circle x² + y² = 1, rotate a ray through angle θ from the positive x-axis and call its circle intersection P. Define cos θ as P's x-coordinate and sin θ as its y-coordinate. The pair order is therefore P = (cos θ, sin θ).

In the first quadrant, dropping a perpendicular to the x-axis forms a right triangle with hypotenuse 1. Coordinates match triangle ratios: adjacent/1 and opposite/1. Thus the new definition extends the old without contradicting it. In other quadrants, coordinates provide signs that leg lengths could not provide.

Quadrants, reference angles and turns

In the first quadrant, sine and cosine are positive. In the second, sine is positive and cosine negative. In the third, both are negative. In the fourth, sine is negative and cosine positive. We can recover these signs by thinking only up/down and left/right, without memorizing a mnemonic sentence.

Angles differing by a whole turn end at the same point. Thus sin(θ + 2kπ) = sin θ and cos(θ + 2kπ) = cos θ for every integer k. For example, 390° and 30° differ by 360° and have the same coordinates. That does not mean the rotations travelled are equal; it means the endpoint coincides.

The reference angle is the acute angle with the horizontal axis when the terminal side is not on an axis. For 150°, the reference is 30°: the point is left and above, so sin 150° = sin 30° and cos 150° = −cos 30°. Reflection across the x-axis gives sin(−θ) = −sin θ and cos(−θ) = cos θ. Signs follow from symmetry.

Build exact values and identities

An isosceles right triangle with legs 1 has hypotenuse √2; hence sin 45° = cos 45° = 1/√2 = √2/2. For 30° and 60°, halve an equilateral triangle of side 2: obtain hypotenuse 2, shorter leg 1 and height √3. Thus sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2 and cos 60° = 1/2.

On the axes, coordinates directly give cos 0 = 1 and sin 0 = 0; cos(π/2) = 0 and sin(π/2) = 1; cos π = −1 and sin π = 0. These values belong to the circle definition without needing a triangle with a positive leg in every direction.

Substituting x = cos θ and y = sin θ into x² + y² = 1 gives cos² θ + sin² θ = 1. Writing sin² θ means (sin θ)², not sin(θ²). Define tan θ = sin θ/cos θ when cos θ ≠ 0. Tangent therefore does not exist at θ = π/2 + kπ for integer k. The identity does not allow signs to be ignored when recovering sine or cosine from a square.

From the unit circle to any radius

If the circle has radius r and center at the origin, an angle-θ point has coordinates (r cos θ, r sin θ). This scales the unit circle by r. For r = 10 m and θ = 60°, we get x = 5 m and y = 5√3 m. Distance to origin confirms √(25 + 75) = 10 m.

Circular-sector area also follows the fraction of a turn: A = (θ/(2π))πr² = r²θ/2, with θ in radians for the final expression. An angle π/3 represents one sixth of a turn, so its arc is one sixth of the circumference and the sector occupies one sixth of the disk area.

For approximate calculations, check the calculator's DEG or RAD mode. sin(30°) = 0,5, but sin(30 rad) is not 0,5. Keep π in exact answers whenever possible; rounding π before finishing introduces unnecessary error. If a quadrant is given, use it to choose the root's sign when applying the fundamental identity.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

Radians are another name for degrees.

They are another measure of the same opening; a turn is 360° or 2π rad.

The circle point is (sine, cosine).

Cartesian order is (cosine, sine).

A positive root always works when recovering cosine.

The quadrant decides whether cosine is positive or negative.

Tangent exists at every angle.

It is a quotient and requires nonzero cosine.